In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales are sometimes referred to as complete residuated semigroups.
Q
\ast\colonQ x Q\toQ
x*\left(veei\in{yi}\right)=veei\in(x*yi)
and
\left(veei\in{yi}\right)*{x}=veei\in(yi*x)
for all
x,yi\inQ
i\inI
I
e
x*e=x=e*x
for all
x\inQ
\ast
A unital quantale may be defined equivalently as a monoid in the category Sup of complete join semi-lattices.
A unital quantale is an idempotent semiring under join and multiplication.
A unital quantale in which the identity is the top element of the underlying lattice is said to be strictly two-sided (or simply integral).
A commutative quantale is a quantale whose multiplication is commutative. A frame, with its multiplication given by the meet operation, is a typical example of a strictly two-sided commutative quantale. Another simple example is provided by the unit interval together with its usual multiplication.
An idempotent quantale is a quantale whose multiplication is idempotent. A frame is the same as an idempotent strictly two-sided quantale.
An involutive quantale is a quantale with an involution
(xy)\circ=y\circx\circ
l(veei\in
| \circ | |
| {x | |
| i}r) |
=veei\in
| \circ). | |
| (x | |
| i |
f\colonQ1\toQ2
x,y,xi\inQ1
i\inI
f(xy)=f(x)f(y),
f\left(veei{xi}\right)=veeif(xi).